LMFDB - Embedded newform 6039.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 $ x^14 - x^13 - 21x^12 + 20x^11 + 167x^10 - 148x^9 - 627x^8 + 497x^7 + 1123x^6 - 745x^5 - 802x^4 + 386x^3 + 74x^2 - 15x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.54331$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54331 q^{2} +4.46845 q^{4} -0.329781 q^{5} +0.595463 q^{7} -6.27804 q^{8} +O(q^{10})$	\(q-2.54331 q^{2} +4.46845 q^{4} -0.329781 q^{5} +0.595463 q^{7} -6.27804 q^{8} +0.838736 q^{10} +1.00000 q^{11} +0.0829823 q^{13} -1.51445 q^{14} +7.03014 q^{16} -6.96049 q^{17} +1.08146 q^{19} -1.47361 q^{20} -2.54331 q^{22} -3.20203 q^{23} -4.89124 q^{25} -0.211050 q^{26} +2.66080 q^{28} -8.72957 q^{29} -3.05943 q^{31} -5.32378 q^{32} +17.7027 q^{34} -0.196372 q^{35} -5.63463 q^{37} -2.75050 q^{38} +2.07038 q^{40} +0.0715670 q^{41} +11.3684 q^{43} +4.46845 q^{44} +8.14377 q^{46} -11.2549 q^{47} -6.64542 q^{49} +12.4400 q^{50} +0.370802 q^{52} +8.24041 q^{53} -0.329781 q^{55} -3.73835 q^{56} +22.2021 q^{58} +0.674261 q^{59} +1.00000 q^{61} +7.78108 q^{62} -0.520247 q^{64} -0.0273660 q^{65} +0.140859 q^{67} -31.1026 q^{68} +0.499437 q^{70} +12.8499 q^{71} -1.10649 q^{73} +14.3306 q^{74} +4.83247 q^{76} +0.595463 q^{77} +14.9245 q^{79} -2.31841 q^{80} -0.182017 q^{82} +1.08794 q^{83} +2.29543 q^{85} -28.9134 q^{86} -6.27804 q^{88} +1.53437 q^{89} +0.0494129 q^{91} -14.3081 q^{92} +28.6246 q^{94} -0.356646 q^{95} -8.74805 q^{97} +16.9014 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.54331	−1.79840	−0.899198	−	0.437543i	$-0.855849\pi$
$2$	−2.54331	−1.79840	−0.899198	+	0.437543i	$0.855849\pi$
$3$	0	0
$3$	0	0
$4$	4.46845	2.23422
$5$	−0.329781	−0.147482	−0.0737412	−	0.997277i	$-0.523494\pi$
$5$	−0.329781	−0.147482	−0.0737412	+	0.997277i	$0.523494\pi$
$6$	0	0
$7$	0.595463	0.225064	0.112532	−	0.993648i	$-0.464104\pi$
$7$	0.595463	0.225064	0.112532	+	0.993648i	$0.464104\pi$
$8$	−6.27804	−2.21962
$9$	0	0
$10$	0.838736	0.265232
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.0829823	0.0230152	0.0115076	−	0.999934i	$-0.496337\pi$
$13$	0.0829823	0.0230152	0.0115076	+	0.999934i	$0.496337\pi$
$14$	−1.51445	−0.404754
$15$	0	0
$16$	7.03014	1.75754
$17$	−6.96049	−1.68817	−0.844083	−	0.536213i	$-0.819854\pi$
$17$	−6.96049	−1.68817	−0.844083	+	0.536213i	$0.819854\pi$
$18$	0	0
$19$	1.08146	0.248105	0.124052	−	0.992276i	$-0.460411\pi$
$19$	1.08146	0.248105	0.124052	+	0.992276i	$0.460411\pi$
$20$	−1.47361	−0.329509
$21$	0	0
$22$	−2.54331	−0.542237
$23$	−3.20203	−0.667669	−0.333835	−	0.942632i	$-0.608343\pi$
$23$	−3.20203	−0.667669	−0.333835	+	0.942632i	$0.608343\pi$
$24$	0	0
$25$	−4.89124	−0.978249
$26$	−0.211050	−0.0413903
$27$	0	0
$28$	2.66080	0.502844
$29$	−8.72957	−1.62104	−0.810521	−	0.585710i	$-0.800816\pi$
$29$	−8.72957	−1.62104	−0.810521	+	0.585710i	$0.800816\pi$
$30$	0	0
$31$	−3.05943	−0.549489	−0.274745	−	0.961517i	$-0.588593\pi$
$31$	−3.05943	−0.549489	−0.274745	+	0.961517i	$0.588593\pi$
$32$	−5.32378	−0.941119
$33$	0	0
$34$	17.7027	3.03599
$35$	−0.196372	−0.0331930
$36$	0	0
$37$	−5.63463	−0.926328	−0.463164	−	0.886273i	$-0.653286\pi$
$37$	−5.63463	−0.926328	−0.463164	+	0.886273i	$0.653286\pi$
$38$	−2.75050	−0.446190
$39$	0	0
$40$	2.07038	0.327355
$41$	0.0715670	0.0111769	0.00558844	−	0.999984i	$-0.498221\pi$
$41$	0.0715670	0.0111769	0.00558844	+	0.999984i	$0.498221\pi$
$42$	0	0
$43$	11.3684	1.73366	0.866831	−	0.498603i	$-0.166153\pi$
$43$	11.3684	1.73366	0.866831	+	0.498603i	$0.166153\pi$
$44$	4.46845	0.673644
$45$	0	0
$46$	8.14377	1.20073
$47$	−11.2549	−1.64169	−0.820845	−	0.571151i	$-0.806497\pi$
$47$	−11.2549	−1.64169	−0.820845	+	0.571151i	$0.806497\pi$
$48$	0	0
$49$	−6.64542	−0.949346
$50$	12.4400	1.75928
$51$	0	0
$52$	0.370802	0.0514210
$53$	8.24041	1.13191	0.565954	−	0.824437i	$-0.308508\pi$
$53$	8.24041	1.13191	0.565954	+	0.824437i	$0.308508\pi$
$54$	0	0
$55$	−0.329781	−0.0444676
$56$	−3.73835	−0.499557
$57$	0	0
$58$	22.2021	2.91527
$59$	0.674261	0.0877813	0.0438907	−	0.999036i	$-0.486025\pi$
$59$	0.674261	0.0877813	0.0438907	+	0.999036i	$0.486025\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	7.78108	0.988199
$63$	0	0
$64$	−0.520247	−0.0650309
$65$	−0.0273660	−0.00339433
$66$	0	0
$67$	0.140859	0.0172087	0.00860433	−	0.999963i	$-0.497261\pi$
$67$	0.140859	0.0172087	0.00860433	+	0.999963i	$0.497261\pi$
$68$	−31.1026	−3.77174
$69$	0	0
$70$	0.499437	0.0596941
$71$	12.8499	1.52500	0.762500	−	0.646988i	$-0.223972\pi$
$71$	12.8499	1.52500	0.762500	+	0.646988i	$0.223972\pi$
$72$	0	0
$73$	−1.10649	−0.129505	−0.0647527	−	0.997901i	$-0.520626\pi$
$73$	−1.10649	−0.129505	−0.0647527	+	0.997901i	$0.520626\pi$
$74$	14.3306	1.66590
$75$	0	0
$76$	4.83247	0.554322
$77$	0.595463	0.0678594
$78$	0	0
$79$	14.9245	1.67913	0.839567	−	0.543256i	$-0.182809\pi$
$79$	14.9245	1.67913	0.839567	+	0.543256i	$0.182809\pi$
$80$	−2.31841	−0.259206
$81$	0	0
$82$	−0.182017	−0.0201005
$83$	1.08794	0.119417	0.0597087	−	0.998216i	$-0.480983\pi$
$83$	1.08794	0.119417	0.0597087	+	0.998216i	$0.480983\pi$
$84$	0	0
$85$	2.29543	0.248975
$86$	−28.9134	−3.11781
$87$	0	0
$88$	−6.27804	−0.669242
$89$	1.53437	0.162642	0.0813212	−	0.996688i	$-0.474086\pi$
$89$	1.53437	0.162642	0.0813212	+	0.996688i	$0.474086\pi$
$90$	0	0
$91$	0.0494129	0.00517988
$92$	−14.3081	−1.49172
$93$	0	0
$94$	28.6246	2.95241
$95$	−0.356646	−0.0365911
$96$	0	0
$97$	−8.74805	−0.888230	−0.444115	−	0.895970i	$-0.646482\pi$
$97$	−8.74805	−0.888230	−0.444115	+	0.895970i	$0.646482\pi$
$98$	16.9014	1.70730
$99$	0	0
$100$	−21.8563	−2.18563
$101$	−9.88085	−0.983181	−0.491590	−	0.870827i	$-0.663584\pi$
$101$	−9.88085	−0.983181	−0.491590	+	0.870827i	$0.663584\pi$
$102$	0	0
$103$	−7.23771	−0.713153	−0.356577	−	0.934266i	$-0.616056\pi$
$103$	−7.23771	−0.713153	−0.356577	+	0.934266i	$0.616056\pi$
$104$	−0.520967	−0.0510850
$105$	0	0
$106$	−20.9580	−2.03562
$107$	16.9166	1.63539	0.817695	−	0.575652i	$-0.195252\pi$
$107$	16.9166	1.63539	0.817695	+	0.575652i	$0.195252\pi$
$108$	0	0
$109$	5.79446	0.555008	0.277504	−	0.960724i	$-0.410493\pi$
$109$	5.79446	0.555008	0.277504	+	0.960724i	$0.410493\pi$
$110$	0.838736	0.0799704
$111$	0	0
$112$	4.18619	0.395558
$113$	12.3271	1.15964	0.579818	−	0.814746i	$-0.303124\pi$
$113$	12.3271	1.15964	0.579818	+	0.814746i	$0.303124\pi$
$114$	0	0
$115$	1.05597	0.0984695
$116$	−39.0077	−3.62177
$117$	0	0
$118$	−1.71486	−0.157865
$119$	−4.14471	−0.379945
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.54331	−0.230261
$123$	0	0
$124$	−13.6709	−1.22768
$125$	3.26194	0.291757
$126$	0	0
$127$	−7.99339	−0.709299	−0.354649	−	0.934999i	$-0.615400\pi$
$127$	−7.99339	−0.709299	−0.354649	+	0.934999i	$0.615400\pi$
$128$	11.9707	1.05807
$129$	0	0
$130$	0.0696003	0.00610435
$131$	15.8726	1.38680	0.693398	−	0.720555i	$-0.256113\pi$
$131$	15.8726	1.38680	0.693398	+	0.720555i	$0.256113\pi$
$132$	0	0
$133$	0.643972	0.0558395
$134$	−0.358249	−0.0309480
$135$	0	0
$136$	43.6982	3.74709
$137$	16.9666	1.44956	0.724778	−	0.688983i	$-0.241942\pi$
$137$	16.9666	1.44956	0.724778	+	0.688983i	$0.241942\pi$
$138$	0	0
$139$	15.7351	1.33463	0.667315	−	0.744775i	$-0.267443\pi$
$139$	15.7351	1.33463	0.667315	+	0.744775i	$0.267443\pi$
$140$	−0.877480	−0.0741606
$141$	0	0
$142$	−32.6813	−2.74255
$143$	0.0829823	0.00693933
$144$	0	0
$145$	2.87885	0.239075
$146$	2.81416	0.232902
$147$	0	0
$148$	−25.1781	−2.06963
$149$	15.3791	1.25990	0.629950	−	0.776635i	$-0.283075\pi$
$149$	15.3791	1.25990	0.629950	+	0.776635i	$0.283075\pi$
$150$	0	0
$151$	16.5708	1.34851	0.674257	−	0.738497i	$-0.264464\pi$
$151$	16.5708	1.34851	0.674257	+	0.738497i	$0.264464\pi$
$152$	−6.78948	−0.550699
$153$	0	0
$154$	−1.51445	−0.122038
$155$	1.00894	0.0810400
$156$	0	0
$157$	−6.06333	−0.483907	−0.241953	−	0.970288i	$-0.577788\pi$
$157$	−6.06333	−0.483907	−0.241953	+	0.970288i	$0.577788\pi$
$158$	−37.9576	−3.01975
$159$	0	0
$160$	1.75568	0.138799
$161$	−1.90669	−0.150268
$162$	0	0
$163$	17.8621	1.39907	0.699535	−	0.714598i	$-0.253390\pi$
$163$	17.8621	1.39907	0.699535	+	0.714598i	$0.253390\pi$
$164$	0.319793	0.0249717
$165$	0	0
$166$	−2.76698	−0.214759
$167$	−21.2052	−1.64091	−0.820455	−	0.571711i	$-0.806280\pi$
$167$	−21.2052	−1.64091	−0.820455	+	0.571711i	$0.806280\pi$
$168$	0	0
$169$	−12.9931	−0.999470
$170$	−5.83801	−0.447755
$171$	0	0
$172$	50.7990	3.87339
$173$	−2.96793	−0.225647	−0.112824	−	0.993615i	$-0.535990\pi$
$173$	−2.96793	−0.225647	−0.112824	+	0.993615i	$0.535990\pi$
$174$	0	0
$175$	−2.91256	−0.220169
$176$	7.03014	0.529917
$177$	0	0
$178$	−3.90237	−0.292495
$179$	−1.16838	−0.0873291	−0.0436646	−	0.999046i	$-0.513903\pi$
$179$	−1.16838	−0.0873291	−0.0436646	+	0.999046i	$0.513903\pi$
$180$	0	0
$181$	−11.0228	−0.819318	−0.409659	−	0.912239i	$-0.634352\pi$
$181$	−11.0228	−0.819318	−0.409659	+	0.912239i	$0.634352\pi$
$182$	−0.125673	−0.00931548
$183$	0	0
$184$	20.1025	1.48197
$185$	1.85819	0.136617
$186$	0	0
$187$	−6.96049	−0.509001
$188$	−50.2918	−3.66790
$189$	0	0
$190$	0.907063	0.0658052
$191$	15.3640	1.11170	0.555849	−	0.831283i	$-0.312393\pi$
$191$	15.3640	1.11170	0.555849	+	0.831283i	$0.312393\pi$
$192$	0	0
$193$	−11.1185	−0.800324	−0.400162	−	0.916444i	$-0.631046\pi$
$193$	−11.1185	−0.800324	−0.400162	+	0.916444i	$0.631046\pi$
$194$	22.2490	1.59739
$195$	0	0
$196$	−29.6947	−2.12105
$197$	17.7144	1.26210	0.631050	−	0.775742i	$-0.282624\pi$
$197$	17.7144	1.26210	0.631050	+	0.775742i	$0.282624\pi$
$198$	0	0
$199$	24.5173	1.73798	0.868992	−	0.494826i	$-0.164768\pi$
$199$	24.5173	1.73798	0.868992	+	0.494826i	$0.164768\pi$
$200$	30.7074	2.17134
$201$	0	0
$202$	25.1301	1.76815
$203$	−5.19814	−0.364838
$204$	0	0
$205$	−0.0236014	−0.00164839
$206$	18.4078	1.28253
$207$	0	0
$208$	0.583378	0.0404500
$209$	1.08146	0.0748064
$210$	0	0
$211$	9.34349	0.643232	0.321616	−	0.946870i	$-0.395774\pi$
$211$	9.34349	0.643232	0.321616	+	0.946870i	$0.395774\pi$
$212$	36.8219	2.52894
$213$	0	0
$214$	−43.0243	−2.94108
$215$	−3.74907	−0.255685
$216$	0	0
$217$	−1.82178	−0.123670
$218$	−14.7371	−0.998124
$219$	0	0
$220$	−1.47361	−0.0993507
$221$	−0.577597	−0.0388534
$222$	0	0
$223$	−23.8528	−1.59730	−0.798652	−	0.601793i	$-0.794453\pi$
$223$	−23.8528	−1.59730	−0.798652	+	0.601793i	$0.794453\pi$
$224$	−3.17011	−0.211812
$225$	0	0
$226$	−31.3517	−2.08548
$227$	−4.98647	−0.330964	−0.165482	−	0.986213i	$-0.552918\pi$
$227$	−4.98647	−0.330964	−0.165482	+	0.986213i	$0.552918\pi$
$228$	0	0
$229$	0.915859	0.0605217	0.0302608	−	0.999542i	$-0.490366\pi$
$229$	0.915859	0.0605217	0.0302608	+	0.999542i	$0.490366\pi$
$230$	−2.68566	−0.177087
$231$	0	0
$232$	54.8046	3.59810
$233$	−7.32305	−0.479749	−0.239875	−	0.970804i	$-0.577106\pi$
$233$	−7.32305	−0.479749	−0.239875	+	0.970804i	$0.577106\pi$
$234$	0	0
$235$	3.71164	0.242120
$236$	3.01290	0.196123
$237$	0	0
$238$	10.5413	0.683292
$239$	24.5442	1.58763	0.793817	−	0.608157i	$-0.208091\pi$
$239$	24.5442	1.58763	0.793817	+	0.608157i	$0.208091\pi$
$240$	0	0
$241$	18.3505	1.18206	0.591031	−	0.806649i	$-0.298721\pi$
$241$	18.3505	1.18206	0.591031	+	0.806649i	$0.298721\pi$
$242$	−2.54331	−0.163490
$243$	0	0
$244$	4.46845	0.286063
$245$	2.19153	0.140012
$246$	0	0
$247$	0.0897424	0.00571017
$248$	19.2072	1.21966
$249$	0	0
$250$	−8.29614	−0.524694
$251$	−2.34673	−0.148125	−0.0740623	−	0.997254i	$-0.523596\pi$
$251$	−2.34673	−0.148125	−0.0740623	+	0.997254i	$0.523596\pi$
$252$	0	0
$253$	−3.20203	−0.201310
$254$	20.3297	1.27560
$255$	0	0
$256$	−29.4048	−1.83780
$257$	2.07933	0.129705	0.0648527	−	0.997895i	$-0.479342\pi$
$257$	2.07933	0.129705	0.0648527	+	0.997895i	$0.479342\pi$
$258$	0	0
$259$	−3.35522	−0.208483
$260$	−0.122283	−0.00758370
$261$	0	0
$262$	−40.3690	−2.49401
$263$	24.1181	1.48718	0.743591	−	0.668634i	$-0.233121\pi$
$263$	24.1181	1.48718	0.743591	+	0.668634i	$0.233121\pi$
$264$	0	0
$265$	−2.71753	−0.166936
$266$	−1.63782	−0.100421
$267$	0	0
$268$	0.629421	0.0384480
$269$	13.9942	0.853243	0.426622	−	0.904430i	$-0.359704\pi$
$269$	13.9942	0.853243	0.426622	+	0.904430i	$0.359704\pi$
$270$	0	0
$271$	−24.2791	−1.47485	−0.737426	−	0.675428i	$-0.763959\pi$
$271$	−24.2791	−1.47485	−0.737426	+	0.675428i	$0.763959\pi$
$272$	−48.9332	−2.96701
$273$	0	0
$274$	−43.1514	−2.60687
$275$	−4.89124	−0.294953
$276$	0	0
$277$	−16.3119	−0.980085	−0.490043	−	0.871699i	$-0.663019\pi$
$277$	−16.3119	−0.980085	−0.490043	+	0.871699i	$0.663019\pi$
$278$	−40.0192	−2.40019
$279$	0	0
$280$	1.23283	0.0736759
$281$	−13.7085	−0.817778	−0.408889	−	0.912584i	$-0.634084\pi$
$281$	−13.7085	−0.817778	−0.408889	+	0.912584i	$0.634084\pi$
$282$	0	0
$283$	31.0223	1.84409	0.922043	−	0.387087i	$-0.126519\pi$
$283$	31.0223	1.84409	0.922043	+	0.387087i	$0.126519\pi$
$284$	57.4190	3.40719
$285$	0	0
$286$	−0.211050	−0.0124797
$287$	0.0426155	0.00251551
$288$	0	0
$289$	31.4484	1.84990
$290$	−7.32181	−0.429951
$291$	0	0
$292$	−4.94431	−0.289344
$293$	−4.76769	−0.278531	−0.139266	−	0.990255i	$-0.544474\pi$
$293$	−4.76769	−0.278531	−0.139266	+	0.990255i	$0.544474\pi$
$294$	0	0
$295$	−0.222358	−0.0129462
$296$	35.3745	2.05610
$297$	0	0
$298$	−39.1138	−2.26580
$299$	−0.265712	−0.0153665
$300$	0	0
$301$	6.76945	0.390185
$302$	−42.1448	−2.42516
$303$	0	0
$304$	7.60284	0.436053
$305$	−0.329781	−0.0188832
$306$	0	0
$307$	−9.09194	−0.518905	−0.259452	−	0.965756i	$-0.583542\pi$
$307$	−9.09194	−0.518905	−0.259452	+	0.965756i	$0.583542\pi$
$308$	2.66080	0.151613
$309$	0	0
$310$	−2.56605	−0.145742
$311$	−0.130155	−0.00738041	−0.00369020	−	0.999993i	$-0.501175\pi$
$311$	−0.130155	−0.00738041	−0.00369020	+	0.999993i	$0.501175\pi$
$312$	0	0
$313$	−18.8073	−1.06305	−0.531526	−	0.847042i	$-0.678381\pi$
$313$	−18.8073	−1.06305	−0.531526	+	0.847042i	$0.678381\pi$
$314$	15.4210	0.870255
$315$	0	0
$316$	66.6892	3.75156
$317$	−21.5314	−1.20932	−0.604661	−	0.796483i	$-0.706692\pi$
$317$	−21.5314	−1.20932	−0.604661	+	0.796483i	$0.706692\pi$
$318$	0	0
$319$	−8.72957	−0.488762
$320$	0.171568	0.00959092
$321$	0	0
$322$	4.84931	0.270242
$323$	−7.52751	−0.418842
$324$	0	0
$325$	−0.405887	−0.0225146
$326$	−45.4290	−2.51608
$327$	0	0
$328$	−0.449301	−0.0248085
$329$	−6.70186	−0.369485
$330$	0	0
$331$	−5.77423	−0.317380	−0.158690	−	0.987328i	$-0.550727\pi$
$331$	−5.77423	−0.317380	−0.158690	+	0.987328i	$0.550727\pi$
$332$	4.86142	0.266805
$333$	0	0
$334$	53.9316	2.95100
$335$	−0.0464526	−0.00253797
$336$	0	0
$337$	−16.2916	−0.887462	−0.443731	−	0.896160i	$-0.646345\pi$
$337$	−16.2916	−0.887462	−0.443731	+	0.896160i	$0.646345\pi$
$338$	33.0456	1.79744
$339$	0	0
$340$	10.2570	0.556266
$341$	−3.05943	−0.165677
$342$	0	0
$343$	−8.12535	−0.438728
$344$	−71.3712	−3.84808
$345$	0	0
$346$	7.54837	0.405803
$347$	18.5188	0.994143	0.497072	−	0.867710i	$-0.334409\pi$
$347$	18.5188	0.994143	0.497072	+	0.867710i	$0.334409\pi$
$348$	0	0
$349$	9.92712	0.531387	0.265693	−	0.964058i	$-0.414399\pi$
$349$	9.92712	0.531387	0.265693	+	0.964058i	$0.414399\pi$
$350$	7.40755	0.395950
$351$	0	0
$352$	−5.32378	−0.283758
$353$	−14.8318	−0.789415	−0.394707	−	0.918807i	$-0.629154\pi$
$353$	−14.8318	−0.789415	−0.394707	+	0.918807i	$0.629154\pi$
$354$	0	0
$355$	−4.23764	−0.224911
$356$	6.85623	0.363380
$357$	0	0
$358$	2.97157	0.157052
$359$	−27.5857	−1.45592	−0.727959	−	0.685621i	$-0.759531\pi$
$359$	−27.5857	−1.45592	−0.727959	+	0.685621i	$0.759531\pi$
$360$	0	0
$361$	−17.8304	−0.938444
$362$	28.0344	1.47346
$363$	0	0
$364$	0.220799	0.0115730
$365$	0.364900	0.0190998
$366$	0	0
$367$	−25.3367	−1.32256	−0.661281	−	0.750138i	$-0.729987\pi$
$367$	−25.3367	−1.32256	−0.661281	+	0.750138i	$0.729987\pi$
$368$	−22.5107	−1.17345
$369$	0	0
$370$	−4.72597	−0.245692
$371$	4.90686	0.254752
$372$	0	0
$373$	22.9549	1.18856	0.594281	−	0.804258i	$-0.297437\pi$
$373$	22.9549	1.18856	0.594281	+	0.804258i	$0.297437\pi$
$374$	17.7027	0.915385
$375$	0	0
$376$	70.6585	3.64393
$377$	−0.724400	−0.0373085
$378$	0	0
$379$	−2.82059	−0.144884	−0.0724420	−	0.997373i	$-0.523079\pi$
$379$	−2.82059	−0.144884	−0.0724420	+	0.997373i	$0.523079\pi$
$380$	−1.59365	−0.0817527
$381$	0	0
$382$	−39.0754	−1.99927
$383$	−25.6130	−1.30876	−0.654382	−	0.756164i	$-0.727071\pi$
$383$	−25.6130	−1.30876	−0.654382	+	0.756164i	$0.727071\pi$
$384$	0	0
$385$	−0.196372	−0.0100081
$386$	28.2777	1.43930
$387$	0	0
$388$	−39.0902	−1.98450
$389$	2.02580	0.102712	0.0513562	−	0.998680i	$-0.483646\pi$
$389$	2.02580	0.102712	0.0513562	+	0.998680i	$0.483646\pi$
$390$	0	0
$391$	22.2877	1.12714
$392$	41.7203	2.10719
$393$	0	0
$394$	−45.0533	−2.26975
$395$	−4.92180	−0.247643
$396$	0	0
$397$	1.07995	0.0542011	0.0271006	−	0.999633i	$-0.491373\pi$
$397$	1.07995	0.0542011	0.0271006	+	0.999633i	$0.491373\pi$
$398$	−62.3552	−3.12558
$399$	0	0
$400$	−34.3861	−1.71931
$401$	−3.50052	−0.174807	−0.0874037	−	0.996173i	$-0.527857\pi$
$401$	−3.50052	−0.174807	−0.0874037	+	0.996173i	$0.527857\pi$
$402$	0	0
$403$	−0.253878	−0.0126466
$404$	−44.1521	−2.19665
$405$	0	0
$406$	13.2205	0.656123
$407$	−5.63463	−0.279298
$408$	0	0
$409$	−27.1693	−1.34344	−0.671719	−	0.740806i	$-0.734444\pi$
$409$	−27.1693	−1.34344	−0.671719	+	0.740806i	$0.734444\pi$
$410$	0.0600258	0.00296446
$411$	0	0
$412$	−32.3414	−1.59334
$413$	0.401498	0.0197564
$414$	0	0
$415$	−0.358783	−0.0176120
$416$	−0.441779	−0.0216600
$417$	0	0
$418$	−2.75050	−0.134531
$419$	15.6757	0.765810	0.382905	−	0.923788i	$-0.374924\pi$
$419$	15.6757	0.765810	0.382905	+	0.923788i	$0.374924\pi$
$420$	0	0
$421$	−1.40707	−0.0685762	−0.0342881	−	0.999412i	$-0.510916\pi$
$421$	−1.40707	−0.0685762	−0.0342881	+	0.999412i	$0.510916\pi$
$422$	−23.7634	−1.15679
$423$	0	0
$424$	−51.7337	−2.51241
$425$	34.0454	1.65145
$426$	0	0
$427$	0.595463	0.0288165
$428$	75.5910	3.65383
$429$	0	0
$430$	9.53507	0.459822
$431$	23.5382	1.13379	0.566897	−	0.823789i	$-0.308144\pi$
$431$	23.5382	1.13379	0.566897	+	0.823789i	$0.308144\pi$
$432$	0	0
$433$	29.2710	1.40668	0.703338	−	0.710856i	$-0.251692\pi$
$433$	29.2710	1.40668	0.703338	+	0.710856i	$0.251692\pi$
$434$	4.63335	0.222408
$435$	0	0
$436$	25.8922	1.24001
$437$	−3.46288	−0.165652
$438$	0	0
$439$	−17.0691	−0.814664	−0.407332	−	0.913280i	$-0.633541\pi$
$439$	−17.0691	−0.814664	−0.407332	+	0.913280i	$0.633541\pi$
$440$	2.07038	0.0987014
$441$	0	0
$442$	1.46901	0.0698738
$443$	32.9257	1.56435	0.782173	−	0.623061i	$-0.214111\pi$
$443$	32.9257	1.56435	0.782173	+	0.623061i	$0.214111\pi$
$444$	0	0
$445$	−0.506004	−0.0239869
$446$	60.6653	2.87258
$447$	0	0
$448$	−0.309788	−0.0146361
$449$	30.7700	1.45213	0.726064	−	0.687627i	$-0.241348\pi$
$449$	30.7700	1.45213	0.726064	+	0.687627i	$0.241348\pi$
$450$	0	0
$451$	0.0715670	0.00336996
$452$	55.0831	2.59089
$453$	0	0
$454$	12.6822	0.595204
$455$	−0.0162954	−0.000763942 0
$456$	0	0
$457$	−11.8994	−0.556632	−0.278316	−	0.960490i	$-0.589776\pi$
$457$	−11.8994	−0.556632	−0.278316	+	0.960490i	$0.589776\pi$
$458$	−2.32932	−0.108842
$459$	0	0
$460$	4.71854	0.220003
$461$	−9.53778	−0.444218	−0.222109	−	0.975022i	$-0.571294\pi$
$461$	−9.53778	−0.444218	−0.222109	+	0.975022i	$0.571294\pi$
$462$	0	0
$463$	22.8038	1.05978	0.529890	−	0.848066i	$-0.322233\pi$
$463$	22.8038	1.05978	0.529890	+	0.848066i	$0.322233\pi$
$464$	−61.3701	−2.84904
$465$	0	0
$466$	18.6248	0.862778
$467$	−41.0776	−1.90084	−0.950422	−	0.310962i	$-0.899349\pi$
$467$	−41.0776	−1.90084	−0.950422	+	0.310962i	$0.899349\pi$
$468$	0	0
$469$	0.0838763	0.00387305
$470$	−9.43986	−0.435428
$471$	0	0
$472$	−4.23304	−0.194842
$473$	11.3684	0.522719
$474$	0	0
$475$	−5.28970	−0.242708
$476$	−18.5204	−0.848883
$477$	0	0
$478$	−62.4237	−2.85519
$479$	11.8369	0.540841	0.270421	−	0.962742i	$-0.412837\pi$
$479$	11.8369	0.540841	0.270421	+	0.962742i	$0.412837\pi$
$480$	0	0
$481$	−0.467575	−0.0213196
$482$	−46.6712	−2.12581
$483$	0	0
$484$	4.46845	0.203111
$485$	2.88494	0.130998
$486$	0	0
$487$	23.2203	1.05221	0.526105	−	0.850420i	$-0.323652\pi$
$487$	23.2203	1.05221	0.526105	+	0.850420i	$0.323652\pi$
$488$	−6.27804	−0.284194
$489$	0	0
$490$	−5.57376	−0.251797
$491$	10.6561	0.480901	0.240451	−	0.970661i	$-0.422705\pi$
$491$	10.6561	0.480901	0.240451	+	0.970661i	$0.422705\pi$
$492$	0	0
$493$	60.7621	2.73659
$494$	−0.228243	−0.0102691
$495$	0	0
$496$	−21.5082	−0.965747
$497$	7.65163	0.343223
$498$	0	0
$499$	41.6152	1.86295	0.931475	−	0.363804i	$-0.118522\pi$
$499$	41.6152	1.86295	0.931475	+	0.363804i	$0.118522\pi$
$500$	14.5758	0.651851
$501$	0	0
$502$	5.96848	0.266386
$503$	−31.7709	−1.41660	−0.708298	−	0.705914i	$-0.750537\pi$
$503$	−31.7709	−1.41660	−0.708298	+	0.705914i	$0.750537\pi$
$504$	0	0
$505$	3.25851	0.145002
$506$	8.14377	0.362035
$507$	0	0
$508$	−35.7181	−1.58473
$509$	−29.3708	−1.30184	−0.650920	−	0.759147i	$-0.725617\pi$
$509$	−29.3708	−1.30184	−0.650920	+	0.759147i	$0.725617\pi$
$510$	0	0
$511$	−0.658877	−0.0291470
$512$	50.8442	2.24702
$513$	0	0
$514$	−5.28840	−0.233261
$515$	2.38686	0.105178
$516$	0	0
$517$	−11.2549	−0.494988
$518$	8.53338	0.374935
$519$	0	0
$520$	0.171805	0.00753414
$521$	3.61714	0.158470	0.0792350	−	0.996856i	$-0.474752\pi$
$521$	3.61714	0.158470	0.0792350	+	0.996856i	$0.474752\pi$
$522$	0	0
$523$	−21.7716	−0.952004	−0.476002	−	0.879444i	$-0.657914\pi$
$523$	−21.7716	−0.952004	−0.476002	+	0.879444i	$0.657914\pi$
$524$	70.9259	3.09841
$525$	0	0
$526$	−61.3398	−2.67454
$527$	21.2951	0.927629
$528$	0	0
$529$	−12.7470	−0.554218
$530$	6.91153	0.300218
$531$	0	0
$532$	2.87756	0.124758
$533$	0.00593879	0.000257238 0
$534$	0	0
$535$	−5.57877	−0.241191
$536$	−0.884319	−0.0381967
$537$	0	0
$538$	−35.5917	−1.53447
$539$	−6.64542	−0.286239
$540$	0	0
$541$	39.5580	1.70073	0.850366	−	0.526191i	$-0.176380\pi$
$541$	39.5580	1.70073	0.850366	+	0.526191i	$0.176380\pi$
$542$	61.7494	2.65236
$543$	0	0
$544$	37.0561	1.58877
$545$	−1.91090	−0.0818540
$546$	0	0
$547$	22.9213	0.980046	0.490023	−	0.871710i	$-0.336988\pi$
$547$	22.9213	0.980046	0.490023	+	0.871710i	$0.336988\pi$
$548$	75.8144	3.23863
$549$	0	0
$550$	12.4400	0.530442
$551$	−9.44072	−0.402188
$552$	0	0
$553$	8.88697	0.377913
$554$	41.4862	1.76258
$555$	0	0
$556$	70.3113	2.98186
$557$	−4.36360	−0.184892	−0.0924458	−	0.995718i	$-0.529468\pi$
$557$	−4.36360	−0.184892	−0.0924458	+	0.995718i	$0.529468\pi$
$558$	0	0
$559$	0.943374	0.0399005
$560$	−1.38053	−0.0583378
$561$	0	0
$562$	34.8649	1.47069
$563$	−7.88133	−0.332158	−0.166079	−	0.986112i	$-0.553111\pi$
$563$	−7.88133	−0.332158	−0.166079	+	0.986112i	$0.553111\pi$
$564$	0	0
$565$	−4.06524	−0.171026
$566$	−78.8995	−3.31640
$567$	0	0
$568$	−80.6721	−3.38493
$569$	−19.5591	−0.819961	−0.409980	−	0.912094i	$-0.634464\pi$
$569$	−19.5591	−0.819961	−0.409980	+	0.912094i	$0.634464\pi$
$570$	0	0
$571$	37.0732	1.55147	0.775733	−	0.631061i	$-0.217380\pi$
$571$	37.0732	1.55147	0.775733	+	0.631061i	$0.217380\pi$
$572$	0.370802	0.0155040
$573$	0	0
$574$	−0.108385	−0.00452389
$575$	15.6619	0.653147
$576$	0	0
$577$	20.2481	0.842941	0.421471	−	0.906842i	$-0.361514\pi$
$577$	20.2481	0.842941	0.421471	+	0.906842i	$0.361514\pi$
$578$	−79.9831	−3.32686
$579$	0	0
$580$	12.8640	0.534147
$581$	0.647830	0.0268765
$582$	0	0
$583$	8.24041	0.341283
$584$	6.94662	0.287453
$585$	0	0
$586$	12.1257	0.500910
$587$	14.4483	0.596343	0.298172	−	0.954512i	$-0.403623\pi$
$587$	14.4483	0.596343	0.298172	+	0.954512i	$0.403623\pi$
$588$	0	0
$589$	−3.30866	−0.136331
$590$	0.565527	0.0232824
$591$	0	0
$592$	−39.6123	−1.62805
$593$	14.6070	0.599839	0.299920	−	0.953964i	$-0.403040\pi$
$593$	14.6070	0.599839	0.299920	+	0.953964i	$0.403040\pi$
$594$	0	0
$595$	1.36685	0.0560353
$596$	68.7205	2.81490
$597$	0	0
$598$	0.675789	0.0276351
$599$	−3.38693	−0.138386	−0.0691932	−	0.997603i	$-0.522042\pi$
$599$	−3.38693	−0.138386	−0.0691932	+	0.997603i	$0.522042\pi$
$600$	0	0
$601$	25.0307	1.02102	0.510512	−	0.859871i	$-0.329456\pi$
$601$	25.0307	1.02102	0.510512	+	0.859871i	$0.329456\pi$
$602$	−17.2168	−0.701706
$603$	0	0
$604$	74.0459	3.01288
$605$	−0.329781	−0.0134075
$606$	0	0
$607$	41.4522	1.68249	0.841246	−	0.540652i	$-0.181822\pi$
$607$	41.4522	1.68249	0.841246	+	0.540652i	$0.181822\pi$
$608$	−5.75747	−0.233496
$609$	0	0
$610$	0.838736	0.0339594
$611$	−0.933954	−0.0377837
$612$	0	0
$613$	−22.4905	−0.908383	−0.454191	−	0.890904i	$-0.650072\pi$
$613$	−22.4905	−0.908383	−0.454191	+	0.890904i	$0.650072\pi$
$614$	23.1237	0.933195
$615$	0	0
$616$	−3.73835	−0.150622
$617$	−19.0197	−0.765703	−0.382851	−	0.923810i	$-0.625058\pi$
$617$	−19.0197	−0.765703	−0.382851	+	0.923810i	$0.625058\pi$
$618$	0	0
$619$	−19.8223	−0.796727	−0.398364	−	0.917228i	$-0.630422\pi$
$619$	−19.8223	−0.796727	−0.398364	+	0.917228i	$0.630422\pi$
$620$	4.50840	0.181062
$621$	0	0
$622$	0.331025	0.0132729
$623$	0.913658	0.0366050
$624$	0	0
$625$	23.3805	0.935220
$626$	47.8329	1.91179
$627$	0	0
$628$	−27.0937	−1.08116
$629$	39.2198	1.56380
$630$	0	0
$631$	−8.96489	−0.356886	−0.178443	−	0.983950i	$-0.557106\pi$
$631$	−8.96489	−0.356886	−0.178443	+	0.983950i	$0.557106\pi$
$632$	−93.6965	−3.72704
$633$	0	0
$634$	54.7611	2.17484
$635$	2.63607	0.104609
$636$	0	0
$637$	−0.551453	−0.0218494
$638$	22.2021	0.878988
$639$	0	0
$640$	−3.94771	−0.156047
$641$	2.96210	0.116996	0.0584980	−	0.998288i	$-0.481369\pi$
$641$	2.96210	0.116996	0.0584980	+	0.998288i	$0.481369\pi$
$642$	0	0
$643$	30.3611	1.19733	0.598663	−	0.801001i	$-0.295699\pi$
$643$	30.3611	1.19733	0.598663	+	0.801001i	$0.295699\pi$
$644$	−8.51995	−0.335733
$645$	0	0
$646$	19.1448	0.753243
$647$	−44.6983	−1.75727	−0.878636	−	0.477491i	$-0.841546\pi$
$647$	−44.6983	−1.75727	−0.878636	+	0.477491i	$0.841546\pi$
$648$	0	0
$649$	0.674261	0.0264671
$650$	1.03230	0.0404901
$651$	0	0
$652$	79.8161	3.12584
$653$	21.4840	0.840733	0.420366	−	0.907355i	$-0.361902\pi$
$653$	21.4840	0.840733	0.420366	+	0.907355i	$0.361902\pi$
$654$	0	0
$655$	−5.23448	−0.204528
$656$	0.503126	0.0196438
$657$	0	0
$658$	17.0449	0.664481
$659$	26.1584	1.01899	0.509493	−	0.860475i	$-0.329833\pi$
$659$	26.1584	1.01899	0.509493	+	0.860475i	$0.329833\pi$
$660$	0	0
$661$	49.2767	1.91664	0.958320	−	0.285697i	$-0.0922249\pi$
$661$	49.2767	1.91664	0.958320	+	0.285697i	$0.0922249\pi$
$662$	14.6857	0.570775
$663$	0	0
$664$	−6.83016	−0.265061
$665$	−0.212370	−0.00823534
$666$	0	0
$667$	27.9523	1.08232
$668$	−94.7545	−3.66616
$669$	0	0
$670$	0.118143	0.00456428
$671$	1.00000	0.0386046
$672$	0	0
$673$	−7.96941	−0.307198	−0.153599	−	0.988133i	$-0.549086\pi$
$673$	−7.96941	−0.307198	−0.153599	+	0.988133i	$0.549086\pi$
$674$	41.4348	1.59601
$675$	0	0
$676$	−58.0591	−2.23304
$677$	−13.0772	−0.502596	−0.251298	−	0.967910i	$-0.580857\pi$
$677$	−13.0772	−0.502596	−0.251298	+	0.967910i	$0.580857\pi$
$678$	0	0
$679$	−5.20914	−0.199909
$680$	−14.4108	−0.552630
$681$	0	0
$682$	7.78108	0.297953
$683$	−29.5060	−1.12902	−0.564508	−	0.825427i	$-0.690934\pi$
$683$	−29.5060	−1.12902	−0.564508	+	0.825427i	$0.690934\pi$
$684$	0	0
$685$	−5.59526	−0.213784
$686$	20.6653	0.789006
$687$	0	0
$688$	79.9213	3.04697
$689$	0.683809	0.0260510
$690$	0	0
$691$	1.06947	0.0406846	0.0203423	−	0.999793i	$-0.493524\pi$
$691$	1.06947	0.0406846	0.0203423	+	0.999793i	$0.493524\pi$
$692$	−13.2620	−0.504147
$693$	0	0
$694$	−47.0992	−1.78786
$695$	−5.18912	−0.196835
$696$	0	0
$697$	−0.498141	−0.0188684
$698$	−25.2478	−0.955643
$699$	0	0
$700$	−13.0146	−0.491906
$701$	2.52697	0.0954424	0.0477212	−	0.998861i	$-0.484804\pi$
$701$	2.52697	0.0954424	0.0477212	+	0.998861i	$0.484804\pi$
$702$	0	0
$703$	−6.09365	−0.229826
$704$	−0.520247	−0.0196076
$705$	0	0
$706$	37.7218	1.41968
$707$	−5.88368	−0.221279
$708$	0	0
$709$	−8.02051	−0.301217	−0.150608	−	0.988594i	$-0.548123\pi$
$709$	−8.02051	−0.301217	−0.150608	+	0.988594i	$0.548123\pi$
$710$	10.7777	0.404478
$711$	0	0
$712$	−9.63281	−0.361005
$713$	9.79637	0.366877
$714$	0	0
$715$	−0.0273660	−0.00102343
$716$	−5.22087	−0.195113
$717$	0	0
$718$	70.1591	2.61831
$719$	17.4383	0.650339	0.325170	−	0.945656i	$-0.394579\pi$
$719$	17.4383	0.650339	0.325170	+	0.945656i	$0.394579\pi$
$720$	0	0
$721$	−4.30979	−0.160505
$722$	45.3484	1.68769
$723$	0	0
$724$	−49.2548	−1.83054
$725$	42.6985	1.58578
$726$	0	0
$727$	−21.4232	−0.794541	−0.397271	−	0.917701i	$-0.630043\pi$
$727$	−21.4232	−0.794541	−0.397271	+	0.917701i	$0.630043\pi$
$728$	−0.310217	−0.0114974
$729$	0	0
$730$	−0.928057	−0.0343489
$731$	−79.1294	−2.92671
$732$	0	0
$733$	24.8791	0.918929	0.459464	−	0.888196i	$-0.348041\pi$
$733$	24.8791	0.918929	0.459464	+	0.888196i	$0.348041\pi$
$734$	64.4391	2.37849
$735$	0	0
$736$	17.0469	0.628356
$737$	0.140859	0.00518861
$738$	0	0
$739$	−3.44825	−0.126846	−0.0634230	−	0.997987i	$-0.520202\pi$
$739$	−3.44825	−0.126846	−0.0634230	+	0.997987i	$0.520202\pi$
$740$	8.30325	0.305233
$741$	0	0
$742$	−12.4797	−0.458144
$743$	18.1232	0.664874	0.332437	−	0.943125i	$-0.392129\pi$
$743$	18.1232	0.664874	0.332437	+	0.943125i	$0.392129\pi$
$744$	0	0
$745$	−5.07171	−0.185813
$746$	−58.3816	−2.13750
$747$	0	0
$748$	−31.1026	−1.13722
$749$	10.0732	0.368067
$750$	0	0
$751$	12.9982	0.474310	0.237155	−	0.971472i	$-0.423785\pi$
$751$	12.9982	0.474310	0.237155	+	0.971472i	$0.423785\pi$
$752$	−79.1233	−2.88533
$753$	0	0
$754$	1.84238	0.0670954
$755$	−5.46474	−0.198882
$756$	0	0
$757$	−16.0448	−0.583158	−0.291579	−	0.956547i	$-0.594181\pi$
$757$	−16.0448	−0.583158	−0.291579	+	0.956547i	$0.594181\pi$
$758$	7.17365	0.260559
$759$	0	0
$760$	2.23904	0.0812185
$761$	−40.4165	−1.46510	−0.732549	−	0.680714i	$-0.761670\pi$
$761$	−40.4165	−1.46510	−0.732549	+	0.680714i	$0.761670\pi$
$762$	0	0
$763$	3.45039	0.124912
$764$	68.6532	2.48379
$765$	0	0
$766$	65.1419	2.35367
$767$	0.0559518	0.00202030
$768$	0	0
$769$	25.1456	0.906774	0.453387	−	0.891314i	$-0.350216\pi$
$769$	25.1456	0.906774	0.453387	+	0.891314i	$0.350216\pi$
$770$	0.499437	0.0179984
$771$	0	0
$772$	−49.6822	−1.78810
$773$	16.3560	0.588284	0.294142	−	0.955762i	$-0.404966\pi$
$773$	16.3560	0.588284	0.294142	+	0.955762i	$0.404966\pi$
$774$	0	0
$775$	14.9644	0.537537
$776$	54.9206	1.97154
$777$	0	0
$778$	−5.15226	−0.184717
$779$	0.0773971	0.00277304
$780$	0	0
$781$	12.8499	0.459805
$782$	−56.6846	−2.02704
$783$	0	0
$784$	−46.7183	−1.66851
$785$	1.99957	0.0713677
$786$	0	0
$787$	28.0304	0.999175	0.499588	−	0.866263i	$-0.333485\pi$
$787$	28.0304	0.999175	0.499588	+	0.866263i	$0.333485\pi$
$788$	79.1560	2.81981
$789$	0	0
$790$	12.5177	0.445359
$791$	7.34034	0.260992
$792$	0	0
$793$	0.0829823	0.00294679
$794$	−2.74665	−0.0974751
$795$	0	0
$796$	109.554	3.88305
$797$	35.6574	1.26305	0.631525	−	0.775356i	$-0.282429\pi$
$797$	35.6574	1.26305	0.631525	+	0.775356i	$0.282429\pi$
$798$	0	0
$799$	78.3393	2.77144
$800$	26.0399	0.920649
$801$	0	0
$802$	8.90292	0.314373
$803$	−1.10649	−0.0390473
$804$	0	0
$805$	0.628790	0.0221619
$806$	0.645692	0.0227435
$807$	0	0
$808$	62.0324	2.18229
$809$	−13.6655	−0.480452	−0.240226	−	0.970717i	$-0.577222\pi$
$809$	−13.6655	−0.480452	−0.240226	+	0.970717i	$0.577222\pi$
$810$	0	0
$811$	47.8791	1.68126	0.840631	−	0.541608i	$-0.182184\pi$
$811$	47.8791	1.68126	0.840631	+	0.541608i	$0.182184\pi$
$812$	−23.2276	−0.815130
$813$	0	0
$814$	14.3306	0.502289
$815$	−5.89059	−0.206338
$816$	0	0
$817$	12.2945	0.430130
$818$	69.1002	2.41603
$819$	0	0
$820$	−0.105462	−0.00368288
$821$	−8.04263	−0.280690	−0.140345	−	0.990103i	$-0.544821\pi$
$821$	−8.04263	−0.280690	−0.140345	+	0.990103i	$0.544821\pi$
$822$	0	0
$823$	12.4433	0.433745	0.216872	−	0.976200i	$-0.430414\pi$
$823$	12.4433	0.433745	0.216872	+	0.976200i	$0.430414\pi$
$824$	45.4387	1.58293
$825$	0	0
$826$	−1.02114	−0.0355298
$827$	−12.2012	−0.424278	−0.212139	−	0.977240i	$-0.568043\pi$
$827$	−12.2012	−0.424278	−0.212139	+	0.977240i	$0.568043\pi$
$828$	0	0
$829$	−21.1707	−0.735288	−0.367644	−	0.929967i	$-0.619835\pi$
$829$	−21.1707	−0.735288	−0.367644	+	0.929967i	$0.619835\pi$
$830$	0.912497	0.0316732
$831$	0	0
$832$	−0.0431713	−0.00149670
$833$	46.2554	1.60265
$834$	0	0
$835$	6.99307	0.242005
$836$	4.83247	0.167134
$837$	0	0
$838$	−39.8683	−1.37723
$839$	39.8488	1.37573	0.687867	−	0.725837i	$-0.258547\pi$
$839$	39.8488	1.37573	0.687867	+	0.725837i	$0.258547\pi$
$840$	0	0
$841$	47.2055	1.62777
$842$	3.57861	0.123327
$843$	0	0
$844$	41.7509	1.43713
$845$	4.28488	0.147404
$846$	0	0
$847$	0.595463	0.0204604
$848$	57.9313	1.98937
$849$	0	0
$850$	−86.5883	−2.96995
$851$	18.0423	0.618481
$852$	0	0
$853$	47.3637	1.62170	0.810851	−	0.585252i	$-0.199005\pi$
$853$	47.3637	1.62170	0.810851	+	0.585252i	$0.199005\pi$
$854$	−1.51445	−0.0518234
$855$	0	0
$856$	−106.203	−3.62995
$857$	−0.0360945	−0.00123297	−0.000616483	−	1.00000i	$-0.500196\pi$
$857$	−0.0360945	−0.00123297	−0.000616483		1.00000i	$0.500196\pi$
$858$	0	0
$859$	−6.64544	−0.226740	−0.113370	−	0.993553i	$-0.536164\pi$
$859$	−6.64544	−0.226740	−0.113370	+	0.993553i	$0.536164\pi$
$860$	−16.7525	−0.571257
$861$	0	0
$862$	−59.8650	−2.03901
$863$	−29.7608	−1.01307	−0.506535	−	0.862219i	$-0.669074\pi$
$863$	−29.7608	−1.01307	−0.506535	+	0.862219i	$0.669074\pi$
$864$	0	0
$865$	0.978765	0.0332790
$866$	−74.4454	−2.52976
$867$	0	0
$868$	−8.14052	−0.276307
$869$	14.9245	0.506278
$870$	0	0
$871$	0.0116888	0.000396060 0
$872$	−36.3779	−1.23191
$873$	0	0
$874$	8.80719	0.297908
$875$	1.94237	0.0656640
$876$	0	0
$877$	−16.7597	−0.565934	−0.282967	−	0.959130i	$-0.591319\pi$
$877$	−16.7597	−0.565934	−0.282967	+	0.959130i	$0.591319\pi$
$878$	43.4121	1.46509
$879$	0	0
$880$	−2.31841	−0.0781534
$881$	−24.3345	−0.819851	−0.409925	−	0.912119i	$-0.634445\pi$
$881$	−24.3345	−0.819851	−0.409925	+	0.912119i	$0.634445\pi$
$882$	0	0
$883$	26.7278	0.899461	0.449731	−	0.893164i	$-0.351520\pi$
$883$	26.7278	0.899461	0.449731	+	0.893164i	$0.351520\pi$
$884$	−2.58096	−0.0868072
$885$	0	0
$886$	−83.7404	−2.81331
$887$	−42.4352	−1.42484	−0.712418	−	0.701756i	$-0.752400\pi$
$887$	−42.4352	−1.42484	−0.712418	+	0.701756i	$0.752400\pi$
$888$	0	0
$889$	−4.75977	−0.159638
$890$	1.28693	0.0431379
$891$	0	0
$892$	−106.585	−3.56874
$893$	−12.1717	−0.407311
$894$	0	0
$895$	0.385311	0.0128795
$896$	7.12812	0.238134
$897$	0	0
$898$	−78.2579	−2.61150
$899$	26.7075	0.890745
$900$	0	0
$901$	−57.3573	−1.91085
$902$	−0.182017	−0.00606052
$903$	0	0
$904$	−77.3901	−2.57396
$905$	3.63511	0.120835
$906$	0	0
$907$	−7.59234	−0.252100	−0.126050	−	0.992024i	$-0.540230\pi$
$907$	−7.59234	−0.252100	−0.126050	+	0.992024i	$0.540230\pi$
$908$	−22.2818	−0.739448
$909$	0	0
$910$	0.0414444	0.00137387
$911$	30.6962	1.01701	0.508505	−	0.861059i	$-0.330198\pi$
$911$	30.6962	1.01701	0.508505	+	0.861059i	$0.330198\pi$
$912$	0	0
$913$	1.08794	0.0360057
$914$	30.2640	1.00104
$915$	0	0
$916$	4.09247	0.135219
$917$	9.45155	0.312118
$918$	0	0
$919$	−47.5570	−1.56876	−0.784380	−	0.620281i	$-0.787019\pi$
$919$	−47.5570	−1.56876	−0.784380	+	0.620281i	$0.787019\pi$
$920$	−6.62941	−0.218565
$921$	0	0
$922$	24.2576	0.798880
$923$	1.06631	0.0350981
$924$	0	0
$925$	27.5604	0.906180
$926$	−57.9972	−1.90590
$927$	0	0
$928$	46.4743	1.52559
$929$	24.6228	0.807847	0.403923	−	0.914793i	$-0.367646\pi$
$929$	24.6228	0.807847	0.403923	+	0.914793i	$0.367646\pi$
$930$	0	0
$931$	−7.18678	−0.235537
$932$	−32.7227	−1.07187
$933$	0	0
$934$	104.473	3.41847
$935$	2.29543	0.0750687
$936$	0	0
$937$	13.4835	0.440487	0.220244	−	0.975445i	$-0.429315\pi$
$937$	13.4835	0.440487	0.220244	+	0.975445i	$0.429315\pi$
$938$	−0.213324	−0.00696527
$939$	0	0
$940$	16.5853	0.540951
$941$	51.2951	1.67217	0.836086	−	0.548598i	$-0.184838\pi$
$941$	51.2951	1.67217	0.836086	+	0.548598i	$0.184838\pi$
$942$	0	0
$943$	−0.229160	−0.00746246
$944$	4.74015	0.154279
$945$	0	0
$946$	−28.9134	−0.940054
$947$	−36.6121	−1.18973	−0.594867	−	0.803824i	$-0.702795\pi$
$947$	−36.6121	−1.18973	−0.594867	+	0.803824i	$0.702795\pi$
$948$	0	0
$949$	−0.0918195	−0.00298059
$950$	13.4534	0.436485
$951$	0	0
$952$	26.0207	0.843336
$953$	34.3089	1.11137	0.555687	−	0.831392i	$-0.312455\pi$
$953$	34.3089	1.11137	0.555687	+	0.831392i	$0.312455\pi$
$954$	0	0
$955$	−5.06675	−0.163956
$956$	109.675	3.54713
$957$	0	0
$958$	−30.1049	−0.972646
$959$	10.1030	0.326243
$960$	0	0
$961$	−21.6399	−0.698062
$962$	1.18919	0.0383410
$963$	0	0
$964$	81.9984	2.64099
$965$	3.66665	0.118034
$966$	0	0
$967$	35.6507	1.14645	0.573225	−	0.819398i	$-0.305692\pi$
$967$	35.6507	1.14645	0.573225	+	0.819398i	$0.305692\pi$
$968$	−6.27804	−0.201784
$969$	0	0
$970$	−7.33730	−0.235587
$971$	−2.45087	−0.0786520	−0.0393260	−	0.999226i	$-0.512521\pi$
$971$	−2.45087	−0.0786520	−0.0393260	+	0.999226i	$0.512521\pi$
$972$	0	0
$973$	9.36965	0.300377
$974$	−59.0564	−1.89229
$975$	0	0
$976$	7.03014	0.225029
$977$	36.7003	1.17415	0.587073	−	0.809534i	$-0.300280\pi$
$977$	36.7003	1.17415	0.587073	+	0.809534i	$0.300280\pi$
$978$	0	0
$979$	1.53437	0.0490385
$980$	9.79275	0.312818
$981$	0	0
$982$	−27.1017	−0.864851
$983$	34.5193	1.10099	0.550497	−	0.834837i	$-0.314438\pi$
$983$	34.5193	1.10099	0.550497	+	0.834837i	$0.314438\pi$
$984$	0	0
$985$	−5.84187	−0.186138
$986$	−154.537	−4.92146
$987$	0	0
$988$	0.401009	0.0127578
$989$	−36.4019	−1.15751
$990$	0	0
$991$	−36.6557	−1.16441	−0.582204	−	0.813043i	$-0.697809\pi$
$991$	−36.6557	−1.16441	−0.582204	+	0.813043i	$0.697809\pi$
$992$	16.2877	0.517135
$993$	0	0
$994$	−19.4605	−0.617250
$995$	−8.08533	−0.256322
$996$	0	0
$997$	17.8981	0.566840	0.283420	−	0.958996i	$-0.408531\pi$
$997$	17.8981	0.566840	0.283420	+	0.958996i	$0.408531\pi$
$998$	−105.840	−3.35032
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.j.1.2		14
3.2	odd	2		2013.2.a.h.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.13	✓	14	3.2	odd	2
6039.2.a.j.1.2		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54331 q^{2} +4.46845 q^{4} -0.329781 q^{5} +0.595463 q^{7} -6.27804 q^{8} +O(q^{10})\)	\(q-2.54331 q^{2} +4.46845 q^{4} -0.329781 q^{5} +0.595463 q^{7} -6.27804 q^{8} +0.838736 q^{10} +1.00000 q^{11} +0.0829823 q^{13} -1.51445 q^{14} +7.03014 q^{16} -6.96049 q^{17} +1.08146 q^{19} -1.47361 q^{20} -2.54331 q^{22} -3.20203 q^{23} -4.89124 q^{25} -0.211050 q^{26} +2.66080 q^{28} -8.72957 q^{29} -3.05943 q^{31} -5.32378 q^{32} +17.7027 q^{34} -0.196372 q^{35} -5.63463 q^{37} -2.75050 q^{38} +2.07038 q^{40} +0.0715670 q^{41} +11.3684 q^{43} +4.46845 q^{44} +8.14377 q^{46} -11.2549 q^{47} -6.64542 q^{49} +12.4400 q^{50} +0.370802 q^{52} +8.24041 q^{53} -0.329781 q^{55} -3.73835 q^{56} +22.2021 q^{58} +0.674261 q^{59} +1.00000 q^{61} +7.78108 q^{62} -0.520247 q^{64} -0.0273660 q^{65} +0.140859 q^{67} -31.1026 q^{68} +0.499437 q^{70} +12.8499 q^{71} -1.10649 q^{73} +14.3306 q^{74} +4.83247 q^{76} +0.595463 q^{77} +14.9245 q^{79} -2.31841 q^{80} -0.182017 q^{82} +1.08794 q^{83} +2.29543 q^{85} -28.9134 q^{86} -6.27804 q^{88} +1.53437 q^{89} +0.0494129 q^{91} -14.3081 q^{92} +28.6246 q^{94} -0.356646 q^{95} -8.74805 q^{97} +16.9014 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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